DOCSLIB.ORG

Continuous Rings with Chain Conditions*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Continuous Rings with Chain Conditions* View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA ELSEVIER Journal of Pure and Applied Algebra 97 (1994) 325 ~332 Continuous rings with chain conditions* W.K. Nicholson”,“, M.F. Yousifb Communicated by C.A. Weibel: received 31 March 1993 Abstract The quasi-Frobenius rings are characterized as the left continuous rings satisfying either (A,) or (Al) and either (S,) or (S,), where these conditions are defined as follows: (A,): ACC on left annihilators; (AZ): R/&X(,$) is left Goldie; (S,): S = r(/(S)) for every minimal right ideal S; and (S,): Every minimal right ideal is essential in a summand of RR. These characterizations extend several results in the literature. In addition, it is shown that, in these rings, Soc(RR) = Soc( RR), Soc(eR) is simple for every primitive idempotent e of R. and there exists a complete set of distinct representatives [Rt,, , Rtn} of the isomorphism classes of the simple left R-modules such that (t, R, , t,R) is a complete set of distinct representatives of the isomorphism classes of the simple right R-modules. 1. Introduction A ring R is called left continuous if every left ideal of R is essential in a direct summand of R and every left ideal of R isomorphic to a direct summand of R is itself a direct summand of R. Continuous rings were introduced by Utumi in [lo] as generalizations of self-injective rings. In [l 11, it was shown by Utumi that a two-sided continuous two-sided Artinian ring is self-injective (and hence quasi-Frobenius). Utumi’s results have been extended by several authors. It is known that if R is a ring such that R has ACC on left annihilators or R/Soc(,R) is left Goldie and either (a) R is left self-injective or (b) R is left and right continuous, then R is quasi-Frobenius (see [l, 2, 4, 61). IVote that there are examples of one-sided continuous two-sided Artinian rings which are not quasi-Frobenius [6]. If R is left self-injective then R is left continuous and A = r(l(A)) for every finitely generated right ideal A. If R is right continuous then, in particular, for every right ideal B of R there exists an idempotent “Supported by NSERC Grant A8075. *Corresponding author. 0022-4049/94;$07.00 (: Elsevier Science B.V. All rights reserved SSDI 0022-4049(94)00017-D 326 W.K. Nicholson. M.F. Yous~~/Journul o/‘Pure and Applied AI&m 97 (1994) 325-332 e in R such that B is essential in the right R-module eR. All the above results can be generalized as follows. Theorem 1. The following stutements ure equicalent,for u ring R: (i) R is quusi-Frobenius. (ii) R is kft continuous with ACC on left annihilators und S = r(l(S)) for etlery minimal right ideal S of R. (iii) R is kft continuous with ACC on left annihilutors und etlery minimul right ideal qf R is essential in eR,ftir some idempotent e in R. (iv) R is kft continuous, R/Soc(,R) is kft Goldie und S = r(l(S)),for ez;erJl minimal right ideul S of R. (v) R is left continuous, R/Soc(,R) is left Goldie und ecery minimal right ideul ofR is essentiul in eR,for some idempotent e in d. Armendariz and Park [2, Theorem 41 showed that, if R is left self-injective and R/SocRR is left Goldie, then R is quasi-Frobenius. This follows immediately from (iv) because left self-injective rings are left continuous and also satisfy rl(1) = I for every finitely generated right ideal I. Moreover, (iii) generalizes a result of Camillo and Yousif [4, Theorem 1] and (v) generalizes a result of Ara and Park [I, Corollary 2.71. A key result in the proof of Theorem 1 is a theorem in which the finiteness conditions are relaxed. More precisely, we show that if R is left continuous, R/Soc(,R) has ACC on left annihilators, and S = rl(S) for every minimal right ideal S of R, then R is a semiprimary ring with Soc( RR) = Soc(R,). We also provide an example of a commutative semiprimary local continuous ring R with .uR = unn(unn(.yR)) for every x E R, but which is not self-injective. Throughout this paper all rings considered are associative with identity and all modules are unitary R-modules. We write J(M), Z(M), E(M), and Sot(M) for the Jacobson radical, the singular submodule, the injective hull and the socle of M, respectively. For any subset X of R, l(X) (resp. r(X)) represents the left (resp. right) annihilator of X in R. The notation A ‘c B will mean RA is essential in KB as left R-modules. For a full account of continuous rings and continuous modules we refer the reader to [S, 10, 111. 2. The results Recall that a ring R is called right Kasch [7] if every simple right R-module is isomorphic to a minimal right ideal of R, equivalently if I(!) # 0 for every proper right ideal I of R. Left Kasch rings are defined analogously. Lemma 1 (Ara and Park [l, Proposition 1.41). [fR is left continuous und SAC is ,jinitely generated und es.sential in RR, then R is u semipe~fkct kft Kusch ring and SOC(~R) c_ Soc(R,). W.K. Nicholson, M.F. Yousif/Journal of’Pure and Applied Algebra 97 (1994) 325-332 327 A submodule N of a module M is said to lie over a summand of M if M = P@Q where P c N and NnQ is small in M. The following result is well known (see [S, Corollary 4.421). Lemma 2. If R is semiperfect, ez;ery kft or right ideal lies over a summand qf R. Lemma 3. Suppose R is semipecfect with R(Soc(RR)) ‘e KR. Then R is a right Kasch riny. Proof. Let T be a maximal right ideal of R. Since R is semiperfect, there exists (by Lemma 2) are idempotent e of R such that eR G T and Tn(1 - e)R is small in R. Since eR 5 T we can write T = eR @( Tn( 1 - e)R). And since Tn( 1 - e)R is small in R, Tn( 1 ~ e)R s J(R), and hence 1(J) E I( T n (1 - e)R). Since SocRR G I(J), it fol- lows by hypothesis that I(Tn(1 - e)R) ? RR. But then I(eR) = R(l - e) # 0 implies that I(T) = I(eR@ Tn( 1 - e)R) = I(eR)nl(T n (1 - e)R) # 0, which completes the proof of Lemma 3. q Lemma 4. Suppose R is a left continuous riny and R/Soc(,R) hus ACC on kji unnihi1utor.s. Then: (i) R is a semiprimary ring. (ii) Soc(Re) is simple ,for every primitice idempotent e of R. (iii) So&R) E Soc(RR). (iv) R is a two-sided Kasch riny. (v) rl( T, n ... n T,) = T, n ... n T,, ,fbr any finite ,family of muximul right ideals T,;.., T, of R. (vi) Ir(L,n ‘.. n L,) = L,n ... n L,, .fk uny ,$nite ,fumily of maximal kft ideuls L,;..,L, ofR. (vii) h(J) = r/(J) = rr(J) = J where J = J(R). (viii) Z(R,) G J(R) = Z(,R). (ix) Z(R,) = I(SocRR) and Z(RR) = r(Soc,R). Proof. (i) By [I, Theorem 1.21. (ii) Since R is semiprimary, SocRR is essential in RR. Now if e is a primitive idempotent and S is a simple submodule of Re then S is essential in a summand of Re. Since Re is indecomposable, Soc(Re) is simple. (iii) This follows from (i), (ii) and Lemma 1. (iv) By Lemma 1, R is a left Kasch ring. That R is right Kasch follows from (i), (iii), and Lemma 3. (v) Since R is right Kasch, T = r/(T) for every maximal right ideal T of R. Now if T1, . , T, is any family of maximal right ideals of R, then /(T,) + ... + /(T,) z I(T,n ..t nT,J. Thus rl(T,n ... n T,) G r(l(T,) + ‘.. + I(T,)) = r/(T,)n ... n rl(T,) = T1 n ... n T,,. Since we always have T,n ... nT, 5 r/(T, I-I ... nT,), the desired result follows. 328 W.K. Nicholson, M.F. Yousif/Journal q/Pure and Applied Algebra 97 (1994) 325-332 (vi) The proof is similar to the one given in (v) above. (vii) Since J is an intersection of a finite family of maximal left (resp. right) ideals we infer, from (v) and (vi) above, that r/(J) = /r(J) = J. Let .K E U(J). Then r(J). x = 0 and hence (Soc,R).u = 0. Since SocRR ‘e RR, .Y E .Z(,R). Since R is left continuous, Z(,R) = J and so .Y E J. Hence U(J) 5 J; the other inclusion holds since J E Z(,R) G r(So&R)) because R is left continuous, and Soc(RR) = r(J) because R is semiprimary. (viii) It is easy to see that if R is any semiperfect ring then Z(R,) G J. For, in this case Z(R,) lies over a summand of R by Lemma 2.
Load more

Recommended publications
  • L-Stability in Rings and Left Quasi-Duo Rings
    University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2018-05-28 L-stability in Rings and Left Quasi-duo Rings Horoub, Ayman Mohammad Abedalqader Horoub, A. M. A, (2018). L-stability in Rings and Left Quasi-duo Rings (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/31958 http://hdl.handle.net/1880/106702 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY L-stability in Rings and Left Quasi-duo Rings by Ayman Mohammad Abedalqader Horoub A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS CALGARY, ALBERTA May, 2018 c Ayman Mohammad Abedalqader Horoub 2018 Abstract A ring R is said to have stable range 1 if, for any element a 2 R and any left ideal L of R, Ra + L = R implies a − u 2 L for some unit u in R. Here we insist only that this holds for all L in some non-empty set L(R) of left ideals of R, and say that R is left L-stable in this case. We say that a class C of rings is affordable if C is the class of left L-stable rings for some left idealtor L.
    [Show full text]
  • On Continuous Rings
    JOURNAL OF ALGEBRA 191, 495]509Ž. 1997 ARTICLE NO. JA966936 On Continuous Rings Mohamed F. Yousif* Department of Mathematics, The Ohio State Uni¨ersity, Lima, Ohio 45804 and Centre de Recerca Matematica, Institut d'Estudis Catalans, Apartat 50, E-08193 Bellaterra, Spain View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by Kent R. Fuller provided by Elsevier - Publisher Connector Received March 11, 1996 We show that if R is a semiperfect ring with essential left socle and rlŽ. K s K for every small right ideal K of R, then R is right continuous. Accordingly some well-known classes of rings, such as dual rings and rings all of whose cyclic right R-modules are essentially embedded in projectives, are shown to be continuous. We also prove that a ring R has a perfect duality if and only if the dual of every simple right R-module is simple and R [ R is a left and right CS-module. In Sect. 2 of the paper we provide a characterization for semiperfect right self-injective rings in terms of the CS-condition. Q 1997 Academic Press According to S. K. Jain and S. Lopez-Permouth wx 15 , a ring R is called a right CEP-ring if every cyclic right R-module is essentially embedded in a projectiveŽ. free right R-module. In a recent and interesting article by J. L. Gomez Pardo and P. A. Guil Asensiowx 9 , right CEP-rings were shown to be right artinian. In this paper we will show that such rings are right continuous, and so R is quasi-Frobenius if and only if MR2Ž.is a right CEP-ring.
    [Show full text]
  • Study of Formal Triangular Matrix Rings
    International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 ISSN 2229-5518 21 STUDY OF FORMAL TRIANGULAR MATRIX RINGS MRS Sunita Salunke SCOE Kharghar Navi Mumbai Email:[email protected] Abstract—There has been a continuous study on formal triangular matrix rings. In this present study 2. Generalities on formal triangular matrix of formal triangular matrix rings few ring theoretic rings characteristic properties of formal triangular matrix rings have been studied in detail. Some definitive re- Proposition II.1. (1) The set of maximal right sults are verified on these rings concerning properties ideals of T is given by such as being respectively left Kasch, right mininjec- tive, clean, potent or a ring of stable rank≤ n. The f(I ⊕ M) ⊕ Kj either I = concepts of a strong left Kasch ring and a strong right A and K is a maximal right ideal of B mininjective ring are introduced and it is determined or I is a maximal right ideal of A and K = when the triangular matrix rings are respectively Bg strong left Kasch or strong right mininjective (2) The set of minimal right ideals of T is the I. INTRODUCTION union of the two sets, fW ⊕ 0j W a simple submodule of (A ⊕ All the rings considered will be associative rings M) g and with identity, all the modules considered will be A f0 ⊕ Kj with 0 the zero submodule of (A ⊕ unital modules. For any ring R the category of right M) and K a minimal right (resp left) R-modules will be denoted by Mod- A ideal of B satisfying KM = 0g: R(resp R-Mod).
    [Show full text]
  • T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules
    Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986 DOI: http://dx.doi.org/10.21123/bsj.2021.18.1.0156 T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules Inaam Mohammed Ali Hadi*1 Shukur Neamah Al-Aeashi 2 Farhan Dakhil Shyaa3 1 Department of Mathematics, College of Education for Pure Science/ Ibn Al-Haitham, University of Baghdad, Iraq. 2 Department of Mathematics, College of Education, University of Kufa, Kufa, Iraq. 3 Department of Mathematics, College of Education, University of Al-Qadisiyah, Qadisiyah, Iraq. *Corresponding author: [email protected] *,[email protected], [email protected] *ORCID ID: https://orcid.org/0000-0002-1058-7410, https://orcid.org/0000-0002-9886-3247, https://orcid.org/0000- 0003-1123-1974 Received 24/2/2019, Accepted 25/4/2020, Published Online First 6/12/2020, Published 1/3/2021 This work is licensed under a Creative Commons Attribution 4.0 International License. Abstract: A new generalizations of coretractable modules are introduced where a module ℳ is called t-essentially (weakly t-essentially) coretractable if for all proper submodule 퐾 of ℳ, there exists f∈End(ℳ), f(퐾)=0 and Imf≤tes ℳ (Im f +퐾 ≤tes ℳ). Some basic properties are studied and many relationships between these classes and other related one are presented. Key words: Coretractable module, Essentially coretractable module, T-essentially coretractable module, Weakly t-essentially coretractable module, Weakly essentially coretractable module. Introduction: In this work, all rings have identity and all weakly essentially coretractable if Imf+퐾 ≤ess ℳ" modules are unitary left R-modules.
    [Show full text]
  • ON SEMIPERFECT F-INJECTIVE RINGS Truong Cong Quynh 1
    International Electronic Journal of Algebra Volume 1 (2007) 18{29 ON SEMIPERFECT F-INJECTIVE RINGS Truong Cong Quynh Received: 12 June 2006; Revised: 26 October 2006 Communicated by W. K. Nicholson Abstract. A ring R is called right F-injective if every right R-homomorphism from a finitely generated right ideal of R to R extends to an endomorphism of R. R is called a right FSE-ring if R is a right F-injective semiperfect ring with essential right socle. The class of right FSE-rings is broader than that of right PF-rings. In this paper, we study and provide some characterizations of this class of rings. We prove that if R is left perfect, right F-injective, then R is QF if and only if R=S is left finitely cogenerated where S = Sr = Sl if and only if R is left semiartinian, Soc2(R) is left finitely generated. It is also proved that R is QF if and only if R is left perfect, mininjective and J2 = r(I) for a finite subset I of R. Some known results are obtained as corollaries. Mathematics Subject Classification (2000): 16D50, 16D70, 16D80 Keywords: F(P)-injective ring, mininjective ring, finitely continuous ring, min-CS, QF-ring, PF-ring, FSE-ring, uniform module 1. Introduction Throughout the paper, R represents an associative ring with identity 1 6= 0 and all modules are unitary R-modules. We write MR (resp., RM) to indicate that M is a right (resp., left) R-module. We also write J (resp., Zr, Sr) for the Jacobson radical (resp., the right singular ideal, the right socle of R) and E(MR) for the injective hull of MR.
    [Show full text]
  • Ideals and Overrings of Divided Domains
    International Electronic Journal of Algebra Volume 8 (2009) 80-113 IDEALS AND OVERRINGS OF DIVIDED DOMAINS Gabriel Picavet Received: 27 November 2009; Revised: 3 June 2010 Communicated by Abdullah Harmanc³ Abstract. New properties of divided domains R are established by looking at multiplicatively closed subsets associated to ring morphisms. Let I be an p ideal of R. We exhibit primary ideals, like I I and In if I is primary. We show that Ass(I) = V(I) \ Spec(RMax(Ass(I))). Moreover, the image of the maximal spectrum of (I : I) is contained in Ass(I). We show that certain intersections of ideals are primary ideals. Goldman prime ideals are prime g- ideals. The characterization of maximal flat epimorphic subextensions gives as a result that R is a valuation subring of PrÄuferhulls. We characterize Fontana- Houston divided ­-domains, divided APVDs and divided PPC-domains. Mathematics Subject Classi¯cation (2000): Primary 13G05; Secondary 13A15, 13B24, 13F05 Keywords: a±ne open subset, almost pseudo-valuation domain, antesharp prime ideal, complete integral closure, conductor overring, divided domain (ring), (flat) epimorphism, fragmented domain, g-ideal ring, G-ideal ring, going-down domain, i-domain, Kaplansky transform, Kasch ring, Manis pair, maximal flat epimorphic (sub)extension, ­-domain, open domain, PPC-domain, power-Ahmes domain, propen domain, primal ideal, primary ideal, treed do- main, unbranched prime ideal, valuation pair. 1. Introduction and notation This paper deals with commutative unital rings and their (homo)morphisms. Dobbs introduced the divided property [7]. Let R be an integral domain with quotient ¯eld K, with R 6= K (i.e.
    [Show full text]
  • 2-Simple Injective Rings
    International Journal of Algebra, Vol. 4, 2010, no. 1, 25 - 37 2-Simple Injective Rings Zhu Zhanmin Department of Mathematics, Jiaxing University Jiaxing, Zhejiang Province, 314001, P.R. China zhanmin [email protected] Chen Jianlong Department of Mathematics, Southeast University Nanjing, Jiangsu Province, 210096, P.R. China [email protected] Abstract A ring R is called right 2-simple injective if, for every 2-generated right ideal I of R, every R-linear map from I to R with simple image extends to R. The class of right 2-simple injective rings is broader than that of right simple injective rings and right 2-injective rings . We study characterizations and properties of right 2-simple injective rings, several conditions under which right 2-simple injective rings are QF -rings are given. Mathematics Subject Classification: 16D50, 16L30, 16L60 Keywords: 2-simple injective rings; Kasch rings; semilocal rings; Semiper- fect rings; QF -rings Throughout this article, R is an associative ring with identity, and all modules are unitary. As usual, J(R)orJ for short, Z(RR)(Z(RR)) and Soc(RR)(Soc(RR)) denote respectively the Jacobson radical, the left (right) singular ideal and the left (right) socle of R. The left annihilators of a subset X of R is denoted by lR(X) (or l(X) for short), and the right annihilators of a subset X of R is denoted by rR(X) (or r(X) for short). If M is an R-module, the notation N ⊆max M means that N is a maximal submodule of M, and we write N ⊆⊕ M if N is a direct summand of M for convenience.
    [Show full text]
  • Retractable and Coretractable Modules
    DOI 10.1007/s10958-016-2705-5 Journal of Mathematical Sciences, Vol. 213, No. 2, February, 2016 RETRACTABLE AND CORETRACTABLE MODULES A. N. Abyzov and A. A. Tuganbaev UDC 512.552 Abstract. In this paper, we study mod-retractable modules, CSL-modules, fully Kasch modules, and their interrelations. Right fully Kasch rings are described. It is proved that for a module M of finite length, the following conditions are equivalent. (1) In the category σ(M), every module is retractable. 1 (2) In the category σ(M), every module is coretractable. (3) M is a CSL-module. (4) ExtR(S1,S2)=0 for any two simple nonisomorphic modules S1,S2 ∈ σ(M). (5) M is a fully Kasch module. 1. Introduction All rings are assumed to be associative and with nonzero identity element; all modules are assumed to be unitary. A module M is said to be retractable if HomR(M,N) = 0 for every nonzero submodule N of M. If every module in the category σ(M) is retractable, then the module M is said to be mod-retractable. A ring R is said to be right mod-retractable if every right R-module is retractable. In [1, Theorem 3.5], it is proved that the class of SV-rings coincides with the class of regular mod-retractable rings R such that every primitive image of R is Artinian. In [1,10], it is proved that the class of commutative mod-retractable rings coincides with the class of commutative semi-Artinian rings. A module M is said to be coretractable if HomR(M/N,M) = 0 for every proper submodule N of the module M.
    [Show full text]
  • Over Goldie-Kerr Commutative RINGS 991
    PROCEEDINGSOF THE AMERICANMATHEMATICAL SOCIETY Volume 120, Number 4, April 1994 POLYNOMIALRINGS OVER GOLDIE-KERRCOMMUTATIVE RINGS CARL FAITH (Communicated by Eric Friedlander) Dedicated to the memory ofPere Menal (1951-1991) Abstract. All rings in this paper are commutative, and ace j. (resp., ace © ) denotes the ace on annihilators (resp., on direct sums of ideals). Any subring of an ace ± ring, e.g., of a Noetherian ring, is an ace ± ring. Together, ace ± and ace © constitute the requirement for a ring to be a Goldie ring. Moreover, a ring R is Goldie iff its classical quotient ring Q is Goldie. A ring R is a Kerr ring (the appellation is for J. Kerr, who in 1990 con- structed the first Goldie rings not Kerr) iff the polynomial ring R[x] has ace ± (in which case R must have ace J.). By the Hilbert Basis theorem, if S is a Noetherian ring, then so is S[x]; hence, any subring R of a Noetherian ring is Kerr. In this note, using results of Levitzki, Herstein, Small, and the author, we show that any Goldie ring R such that Q = QC(R) has nil Jacobson radical (equivalently, the nil radical of R is an intersection of associated prime ideals) is Kerr in a very strong sense: Q is Artinian and, hence, Noetherian (The- orems 1.1 and 2.2). As a corollary we prove that any Goldie ring A that is algebraic over a field k is Artinian, and, hence, any order R in A is a Kerr ring (Theorem 2.5 and Corollary 2.6).
    [Show full text]
  • Kasch Bimodules Rendiconti Del Seminario Matematico Della Università Di Padova, Tome 85 (1991), P
    RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA D. N. DIKRANJAN E. GREGORIO A. ORSATTI Kasch bimodules Rendiconti del Seminario Matematico della Università di Padova, tome 85 (1991), p. 147-160 <http://www.numdam.org/item?id=RSMUP_1991__85__147_0> © Rendiconti del Seminario Matematico della Università di Padova, 1991, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Kasch Bimodules. D. N. DIKRANJAN - E. GREGORIO - A. ORSATTI (*) 0. Introduction. Let A and R be two rings with a non zero identity. Denote by 5lA the full subcategory of Mod-A consisting of all submodules of the finitely generated modules in Mod-A and by Rsl the analogous subcategory of R-Mod. Observe that 5lA and RS are finitely closed. Let A be a Kasch ring, i. e. a ring with a perfect duality [K]. It is easy to check that the functors HomA (-, AA) and HomA (-, AA) induce a duality between 5lA and AM On the other hand, by well known results of Azumaya [Az], if AA and RR are artinian, then every duality between two finitely closed subcategories of Mod-A and R-Mod, both containing all finitely gener- ated modules, induces a duality between TA and RIT- This paper is devoted to the study of the bimodules RKA with the property that the functors HomA ( -, KA ) and HomR (-, RK) give rise to a duality between SA and Such a bimodule will be called a Kasch bi- Since ~’A and Rs are finitely closed subcategories of Mod-A and R-Mod respectively, the above duality is a special case of Morita duali- ty, thus RKA is a Morita bimodule, i.
    [Show full text]
  • A GENERALIZATION of SIMPLE-INJECTIVE RINGS Zhu
    International Electronic Journal of Algebra Volume 26 (2019) 76-86 DOI: 10.24330/ieja.586952 A GENERALIZATION OF SIMPLE-INJECTIVE RINGS Zhu Zhanmin Received: 16 October 2018; Accepted: 28 March 2019 Communicated by Burcu Ung¨or¨ Abstract. A ring R is called right 2-simple J-injective if, for every 2-generated right ideal I ⊆ J(R), every R-linear map from I to R with simple image ex- tends to R. The class of right 2-simple J-injective rings is broader than that of right 2-simple injective rings and right simple J-injective rings. Right 2-simple J-injective right Kasch rings are studied, several conditions under which right 2-simple J-injective rings are QF -rings are given. Mathematics Subject Classification (2010): 16D50, 16L30, 16L60 Keywords: 2-Simple J-injective ring, Kasch ring, semilocal ring, semiperfect ring, QF-ring 1. Introduction Throughout this paper, R is an associative ring with identity, m is a positive integer unless otherwise stated, and all modules are unitary. As usual, J(R) or J for short, Zl (Zr) and Sl (Sr) denote respectively the Jacobson radical, the left (right) singular ideal and the left (right) socle of R. The left annihilator of a subset X of R is denoted by l(X), and the right annihilator of X is denoted by r(X). If M is an R-module, then the notation N ⊆max M means that N is a maximal submodule of M, and the notation N E M means that N is an essential submodule of M. Recall that a ring R is called right simple injective [5] if for every right ideal I of R, every R-linear map γ : I ! R with γ(I) simple extends to R.
    [Show full text]
  • On Pseudo-Frobenius Rings
    Canad. Math. Bull. Vol. 48 (2), 2005 pp. 317–320 On Pseudo-Frobenius Rings Mohamed F. Yousif, Yiqiang Zhou and Nasr Zeyada Abstract. It is proved here that a ring R is right pseudo-Frobenius if and only if R is a right Kasch ring such that the second right singular ideal is injective. Throughout, rings R are associative with an identity and modules are unitary R-modules. We write J(M) and soc(M) for the Jacobson radical and the socle of the module M, respectively. For a submodule N of M, N ≤e M means that N is es- sential in M. The singular submodule of the module M is defined by Z(M) = {m ∈ M : ∃ IR ≤e RR such that mI = 0}. Foraring R, Z(RR) and Z(RR) are called the right singular ideal and the left singular ideal of R, respectively. The second right singular r r ideal of R, denoted by Z2, is defined by the equality Z2/Z(RR) = Z(RR/Z(RR)) and l the second left singular ideal Z2 of R can be defined analogously. We write J(R), Sr, Sl for the Jacobson radical, right socle and left socle of R, respectively. A ring R is called right pseudo-Frobenius, briefly right PF, if every faithful right R-module is a genera- tor of the category of all right R-modules; and the ring R is called right Kasch if every simple right R-module embeds in RR. Analogously, one defines left PF and left Kasch rings. It is a well-known result of Osofsky [6] that R is right PF if and only if R is semiperfect, right self-injective with (Sr)R ≤e RR if and only if R is right Kasch, right r self-injective (see [1]).
    [Show full text]